Topological susceptibility to the one-loop order in chiral perturbation theory

Abstract

We derive the topological susceptibility to the one-loop order in the chiral effective theory of QCD, for an arbitrary number of flavors.

Topological susceptibility on the lattice and the three-flavour quark condensate

“…[35] in the isospin limit, as expected. However, this result as it stands does not exploit the Ward identity eq.…”

“…[14] to obtain the leading-order expression) or by considering only the effective potential for constant sources and determining its dependence on the vacuum angle θ (like the derivation of the NLO expression in ref. [35]). In the latter case, we have performed the calculation keeping all orders in strong isospin-breaking (contrary to ref.…”

“…In the latter case, we have performed the calculation keeping all orders in strong isospin-breaking (contrary to ref. [35]) and obtained in the χPT framework [14]:…”

“…The next-to-leading (NLO) expression for the topological susceptibility was recently computed in ref. [35], by determining the dependence of the χPT potential with respect to the vacuum angle (i.e., taking the source term θ(x) as a constant) and performing the corresponding partial derivatives. The LO formula (1.7) was exploited by the TWQCD collaboration to extract the value of the three-flavour quark condensate [28] from RBC/UKQCD configurations with 2+1 domain-wall fermions [36]: Σ(3, 2 GeV) = [259(6)(9) MeV] 3 (in the M S scheme).…”

Chiral perturbation theory in aθvacuum

“…1 For the N f = 2 case, a nonperturbative expression of the vacuum energy density is known [20], from which one can read off χ t , c 4 [21], and any higher order coefficients c 2n 's. 2 It of course gives contributions at NLO.…”